Finite size effects and scaling properties of charge fluctuations

TL;DR

We study finite-size effects in a schematic quark-meson–type model to understand how finite spatial extent modifies the phase diagram and charge fluctuations near a critical endpoint (CEP) and along a first-order line. The approach replaces the full functional integral by a size-dependent weighting of a single mode, $Z=\int d\bar{φ}\,e^{-βV\mathcal{U}_{\mathrm{eff}}(\bar{φ})}$, and emphasizes finite-volume scaling forms, e.g. $f_s(τ,η)=L^{-d}\mathcal{F}(τL^{1/ν},ηL^{βδ/ν})$ with an effective MF exponent $\tilde{ν}=2/3$. We find a data-collapse of the singular part near the CEP for $L\gtrsim 20$ fm and strong finite-size enhancement of fluctuations at the first-order transition with $χ\propto L^d$ due to phase coexistence; the Binder cumulant $κ_B=χ_4/(βL^dχ_2^2)$ provides a universal way to locate the CEP, while along the freeze-out line the kurtosis ratio $R_{42}=χ_4^B/χ_2^B$ remains largely size-independent far from scaling and shifts with $L$ when the line approaches the phase boundary. The framework offers a qualitative bridge to heavy-ion phenomenology and lattice results and points to extensions to canonical ensembles and analyses of Lee–Yang zeros.

Abstract

An effective model is introduced to illustrate finite volume effects beyond the usual momentum space constraints. The fluctuations of the chiral order parameter and the net baryon number, as well as their scaling properties, are investigated near a critical point and a first-order transition in finite size systems. The finite volume effects on the kurtosis are shown in detail along the approximate freeze-out line. Several implications of the finite volume in the charge fluctuations are discussed.

Figures (12)

• Figure 1: The $P(\phi)=e^{-S_E(\phi)}/\mathcal{Z}$ weights as a function of the field $\phi$ for different (rather small) system sizes close below the first-order phase boundary.
• Figure 2: The chiral condensate $\langle\phi\rangle$ at $\mu=0$ for different system sizes and in the thermodynamic limit.
• Figure 3: The chiral condensate $\langle\phi\rangle$ at a first-order transition for different system sizes and in the thermodynamic limit. The dashed curve shows the $L\to\infty$ meta- and unstable solutions.
• Figure 4: The $L=30,\,32,\,34,\ldots,\,100$ fm results collapsing to a single curve for the singular part of the chiral condensate (top) and the chiral susceptibility (bottom) as a function of the scaled subtracted temperature.
• Figure 5: The scaled chiral susceptibility as a function of the scaled temperature in the neighborhood of the first-order transition.